TAILORING OF COMPOSITES

PERIODIGA POLYTEGH.'lIGA SER. GHEi'>!. ENG. VOL. 42, NO. 1, PP. 55-64 (1998)

THERMODYNAMICS OF COMPLEX SYSTEMS:

SPECIAL PROBLEMS OF COUPLED THERMAL AND MOISTURE FIELDS AND APPLICATION TO

TAILORING OF COMPOSITES

A. SZEKERESl, R. A. HELLER2t

1 Department of Applied Mechanics, Technical University of Budapest, Research Group of Continuum Mechanics,

Hungarian Academy of Sciences.

H-1,521 Budapest, Hungary

2Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University

Blacksburg, VA, 24061 Received: March. 26, 1997

Abstract

In case of composites it is possible to increase the effectiveness of tailoring by involving new parameters, but utilizing special symmetries the enormous increase of needed numerical values can be avoided. Starting with the basic equations of thermo-hygro materials the special features and parameters are shown. Finally, some practical applications to the tailoring of fiber reinforced composites are displayed.

Keywords: thermo-hygro fields, Onsager relations, composites, tailoring

1. Introduction

The effectiveness of tailoring of composite materials requires the involvement of increasing number of parameters into the tailoring procedure. On the other hand, this growing number creates difficulties because to obtain the values of parameters is time consuming and very often a formidable task.

One possible solution of this problem is the following: broadening the possibility of tailoring by involving new parameters, but utilizing spe- cial symmetries the enormous increase of needed numerical values can be avoided. Such symmetries are well known in mechanics and thermodynam- ics, e.g., symmetry of stress tensor due to Ne'wton's axioms; symmetry of strain tensor according to geometry; symmetry of stiffness matrix as a con- sequence of the two previous ones; symmetry of diffusivity and relaxation time matrices due to Onsager's reciprocal relations [1], [2]; symmetry of IBased on the lecture given at Minisymposium on Non-Linear Thermodynamics and Reciprocal Relations, September 22-2,5, 1996. Balatonvihigos, Hungary.

56 A .. SZEKERES and R. A. HELLER

thermal and moisture expansion coefficient matrices based on the thermo- hygro-elastic (THE) constitutive equation (generalized Hooke's law) [6].

Our purpose was, starting with the general questions of coupled fields, to introduce the basic relations of thermo-hygro materials including the extension according to the second sound phenomenon [5]; to emphasize the features of thermo-hygro coupling and searching for the possible reasons of non-symmetry in coupled fields. Finally, the theoretical results were applied to tailoring of composite materials.

F,ig. 1 shows a summary of the most often appearing fields and their couplings in mechanics of solids. Some of them are well known, others are in a fast developing process.

I

Displacement field

I

Thermoelasticity (TE)

Displacement grad.--4~-.J--o. __ Thermo- hygro-elasticity (THE) Velocity gradient ----c~.l:.

--,....()...- Magnetohydrodynamics

I

Temperature

-.!~':'@ __

^..JJ

I

Electromagnetic field

I

Electric

---0'

Magnetic - - - 0 -

I

Moisture conc.

fiel~:

In case of composites

<+~E

Piezoelectricity

Magnetostriction

<

Stress field Photoelasticity ? .

LIght

<

Displacement Strain gage?

Electric

Heat conduction and moisture diffusion (3 X) -' Analogy (fourier, fick)

-Cross-coupling (dufour, soret) - Degradation in composites

Fig. 1. Most frequent coupled fields of mechanics

E.g., thermo-hygro fields, always coupled with displacement fields, of- ten occur in composite materials. In case of tailoring such materials the properties, especially the number thereof, have an important role.

Heat conduction and moisture diffusion have triple coupling:

- analogy between Fourier's and Fick's Laws,

- cross-coupling according to Dufour and Soret effects, - similar effect on degradation in composites.

Let us see the basic equations of thermo-hygro materials [5].

THERMOD}'NAMICS OF COMPLEX SYSTEMS 57

2. Basic Problems of Thermo-Hygro Materials

The independent variable of thermo-hygro materials must satisfy the con- servation laws

(2.1) and constitutive laws

(2.2) Here

1-

^{c and (Yc} stand for the flux and source terms, respectively, corre- sponding to C; PI and a^C are the coefficients; a dot denotes the material time derivative and V the space derivative. We apply these equations for moisture and temperature. Further considerations of this section are based on [5]. (i) Moisture and Fick's Law

Replacing the thermal variables and coefficients of Eqs. (2.1) and (2.2) with their equivalents in terms of moisture: C

=

m the moisture con- centration at a certain point of a solid;

1- = f

the moisture flux, PI

=

1, (Y

=

0 and a

=

Dm the moisture diffusion coefficient. 'With these notations, Eqs.

(2.1), (2.2) read

V·

[+

rh

=

0, [=-Dm

vm .

Substituting (2.4) into (2.3) we obtain

rn =

Dm

v

²m

(2.3) (2.4)

(2.5) Eqs. (2.4) and (2.5) are called Fick's First and Second La\vs. (ii) Tempera- ture and Fourier's Law Let C

=

T that is the temperature at a certain point of a solid. The other notations are then:

1- =

q is the heat flux, PI

=

^pCp ,

where c_p is the specific heat with respect to the ~olume and P is the density;

as above, (Y is neglected and a

=

k is the coefficient of heat conduction.

Equation (2.1) emphasises then the conservation of energy

(2.6) while Eq. (2.2) is the celebrated Fourier's Law

:I

=

^{-kvT (2.7)}

SUbstituting (2.7) into (2.6), the heat conduction equation yields

. 2

T

=

^DTV T. ^(2.8)

Here Dr

=

^{kj pCp.}

The similarity between the final results (2.5) and (2.8) is obvious, as well as between the (first) Fick's Law and the Fourier Law. It means actu- ally that theoretical, numerical, and experimental problems of moisture and temperature diffusion can be treated in a similar way.

•

58 A.SZEKERESand R.A.HELLER

An interesting question arises about the simultaneous transport of moisture and temperature: are the effects caused by these phenomena cou- pled or not? The answer is given in the following section. (iii)

between heat and moisture transport There is experimental evidence that temperature field affects the moisture transport, and vice versa, m9isture concentration affects the temperature field. First, the heat flux, caused by moisture concentration gradient

is associated with the Dufour effect, while the corresponding flux is called the Dufour flux. Secondly, the moisture diffusion due to temperature gradient

L =

^-QmT

vT

is associated with the Soret effect with the corresponding flux being Soret flux. Here

QT'

Q~ are the coupling diffusion coefficients:

thermo and thermal-diffusion coefficients, respectively. Suppose we Dufour and Soret fluxes to our basic equations in the previous section.

coupled fluxes can then be written in the following form

and the coupled final equations (provided Qij s are constant)

where the summation convention over the repeated indices is used.

1,2. The notation is obvious:

1.

=

^{j, 1.2}

=

^q;

Cl

=

^{ni, C2}

= T;

Qll

=

^{Qm, Q12}

=

^QT m' Q21

=

^Qm _T' ^Q22

=

^k·

^,

Dll D12 T

D21

am

D22 ~

=

^am,

=

^{Qm' ....L} pcp' pCp

It should be noted that Eq. (2.12) is the system of coupled equations following the principle of equipresence.

relation for Qij can be added

THERMODYNAMICS OF COMPLEX SYSTEMS 59

3. Special Features of Thermo-Hygro Coupling

The paradox of the heat conduction equation (2.8) is ... vell known. Because of its parabolic character (2.8) predicts infinite speed of heat propagation which contradicts physical principles. Maxwell has pointed out the solution to this naradox but contemporary understanding is based on ideas of Vernotte and Cattaneo who introduced relaxation time for the heat flux into the basic equation (2.7).

The basic (more or less formal) solution to this paradox is the following.

Instead of Eq. (2.7) we use

~

+

^TTR

=

-kY'T. ^(3.1)

where TT is the relaxation time. Introducing (3.1) into (2.6). we obtain

. " 2

T

+

ryT = DTY' T (3.2)

which is a hyperbolic equation, describing wave motion with a finite speed

eT

=

(DT/7T) 1/2 . (3.3)

The striking analogy between the basic models of moisture and heat trans- port (Eqs. (2.5), (2.8)) leads logically to formal generalizations.

First, we are tempted to generalize the constitutive law (2.2)

(3.4) where ^Te is a certain relaxation time.

Let us use this idea, at least formally, to generalize Eq. (2.4), i.e., moisture diffusion. Then, instead of Eq. (2.4) we have

L +

^{T m}

i = -

D m Ym (3.5) and instead of Eq. (2.5)

. .. D ,,2

In

+

^Tmm

=

^{m ,,I In.} (3.6) What is the physical meaning of this equation? Certainly the paradox of the infinite speed is removed, moisture transport is described by a hyperbolic equation, however, the physical mechanism behind it is still not clear 1 neither are the real values of 7m .

a vVe apply now a formal generalization of basic equations including the 'Dufour and Soret effects. First we introduce the relaxation time into the

constitutive law. Then for fluxes i, we have

(3.7)

•

60 A. SZEKERES and R. A. HELLER

Substituting (3.7) into the conservation law, \'le obtain

(3.8) In our case i,j

=

^{1,2 with Cl}

=

^{m, C2}

=

T. The corresponding eq~ations

in these terms are the following.

(i) Constitutive laws

[ f] _{~ +}

^[711 721

J[fl -[ ] [

^{Vm ]}

^{vT '}

^(3.9)

(ii) Governing equations

[ ; ] + [~~~ ~~~] ^{[ ; ]}

~~~ ^{] [} ~~; ^{] + [} :~ ^{] +} [~~~ ~~~] ^[ :~ ^]

^(3.10)

These are the formal results and some comments are in order. It is obvious that 711

=

^{7m and 722}

=

^TT. The other two relaxation times 712

=

^{7;;; and}

721

= 7T

characterize coupling effects. Due to Onsager's reciprocal relations we may suppose

(3.11) Though there is total analogy between Fourier's heat conduction and Fick's moisture diffusion law, there are also basic differences between the two problems [7].

The first one is that heat conduction is in connection with the mJ- crostructure of a single material while moisture diffusion connects two ma- terials (porous solid and fluid), and their macrostructure has a basic role.

As a consequence, dealing with this process one has to take into account the conservation of mass and also inertia terms. Of course, this behavior is present during the coupled processes, too.

Another characteristic difference between heat conduction and cross coupled heat and moisture diffusion is the following: In a heat '-V_'H_~U'­

tion process, there is practically no temperature limitation. In the coupled -process there exist two limitations: freezing and evaporation moisture. This means that also phase changes have to be considered. If the temperature is too low, the fluid freezes and the voids disappear;

porous material the whole problem becomes different. If the temperature too high the fluid begins to evaporate, the pressure rises and after a while, the fluid disappears; without moisture the problem is again different. It apparent that the temperature range of the cross-coupled problem is "tT'lrtl\'-,~

limited. Of course, the whole argument is valid in the case of elastic mc)tJ,:m·<i

THERMODYNAMICS OF COMPLEX SYSTEMS 61

4. Possible Reasons for the Non-Symmetry in Coupled Fields Starting with the basic equations of conventional thermo-elasticity, after derivation, holding as basic functions temperature (T) and displacement (ud one gets the so-called generalized equation of motion and heat conduction:

J1Ui,jj

+

(J1

+ ,\)

uk,ki

=

PUi - pbi - ,BT,i, (4.1 )

pc . 1

T,ii - TT -

j3

^T

o

kUi,i

=

^{O. ( 4.2)}

Formally the same relations are valid in the case of hygro-elasticity. The unsymmetry in coupling is obvious. While the temperature field influences the displacement field through its gradient (T,i) the influence of the displace- ment field on the temperature distribution is accomplished by the divergence and the time rate of the displacement (ui,d.

The question could be regarded from different points of view.

In the general equation of motion (4.1) the role of the temperature field is similar to the body force. (See the second and third terms on the right-hand side). In the general equation of heat conduction (4.2) the role of displacement divergence is similar to the temperature.

[

PC

j3] .

T.·· _,ll

= -.

k ^T

+

T,o-u" k ^I,) (4.3) From another point of view the problem is similar to the constitutive equa- tion of a viscous material, where the stress does not depend on the dis- placement gradient (Ui.j) as in the case of an elastic material, but on the velocity gradient (Vi,j)' Recognition of this feature may lead to a possible approach to the problem. Rheological models give a possible handling of the phenomenon by using the generalized equations.

A possible approach to the reasons for the problem may rely on the physical background, i.e., the reversibility-irreversibility in connection \vith the changes of mechanical work to heat and vice versa. Another possibility, in connection with the previous remarKs, is the structure of the material.

While the displacement is related to the macrostructure of the material, heat conduction (change of temperature) involves the microstructure.

It is also possible that the law of heat conduction is the reason for the problem. Using the modified basic equation, which leads to a hyperbolic equation, one may get a better understanding of the unsymmetry. This consideration may lead to another type of coupled system which is closer to the symmetric one.

There are also several questions from a technical point of view. The role of time (time rate of displacement divergence) has been mentioned, but the speed of changes has not been dealt with. It is possible that in the case of low speed the time rate is negligible [3].

Another technical problem arises from the fact that mathematically the differentiation and integration are not reversible.

62

..

^",

N§

~

$2 ~

~ :~

'"

.z ""

"0

"0

E lii

.r:;

I -

30

25

20

15

10

5 0

A. SZEKERES and R. A. HELLER

Thermal diffusivity (k) v.s.angle

e

t:. Experimental o Calculated

Angle,

e

(degrees J

Fig. 2. Comparison of experimental and calculated values of thermal diffusivity for graphite-epoxy as a function of fiber orientation, at room temperature

~ 34

«J

•

5 33

"0

~ 32

~ E 31

30 29

28 27 26

Temperature at point x = 20 mm

Theory, moisture 0°',

Test, moisture 5

°'0

Test I moisture 8 0'0

25b.1!' ... ~~

24~--~~~~--~~--~~~~--~~--~~~~.

o 100 200 700 800

Fig. 3. Comparison of experimental and theoretical results Time,s

Temperature response 20 mm from a heated end

5. Application for Tailoring of Composite Materials

As an application of the equations introduced in the previous sections, they may be used for the tailoring of composites.

Considering the constitutive equations of anisotropic THE solid, the generalized constitutive law is recalled together with Fourier's heat conduc-

THERMODYNAMICS OF COMPLEX SYSTEMS 63

tion law and Fick's moisture transport relation:

tij

=

^aijkWkI

+

o:i]Y'm, (5.1 )

Instead of the latter two equations, the coupled and modified relation based on the "second sound" phenomenon will be applied. For anisotropic THE materials, Eq. (2.7) becomes

with

Here

Jo:i =

0:,/3=1,2, k,l,i,j = 1,2,3.

(d~)ij], Cj3 ..

⁼

[m,j]

k·· ₁₎ .) T· _,)

(5.2)

are the extended matrices of flux, relaxation, diffusivity, and concentration gradients, respectively.

On the basis of parameters contained in the coefficients of Eqs. (5.1,2) listed in the Introduction, tailoring can be performed. As it was mentioned before, the number of required values can be reduced through various sym- metry conditions. Most of the material properties are available in hand- books though some are not. As applications of the theory, some examples are presented [4, 7] in Figs. 2-5.

c 0.20 .2

Ul c a a.

~ 0.15 0.10 0.05

o

A

Graphite/epoxy

o

^~60

Fiber angle I

e

(degrees)

Fig.

4.

Longitudinal eT/ME of Symmetric Balanced Laminate

64 A.SZEKERESand R.A.HELLER

~T~:~=:::~m_mt::· ~

I ~

Bimaterial tany type: therm . .moist.,th/moi)

Thermal expansion } Needed max.deflection, Moisture expansion i.e. max./min.coeff.

Therm/moist expansion See num.calculations E.g.: in Africa: CTE the best because temp.

diff. day and night in Lappland: CME the best because no

sunshine in between: CT/ME ...

Fig. 5. Light-house energy supply

6. Conclusions

Onsager's reciprocal relations help to define the material properties for coupled fields, e.g., diffusivities and relaxation parameters. In the case of thermo-hygro materials, in the application of reciprocal relations some specialities have to be considered (see dimensions). 'vVe found a possible interpretation for the non-symmetry in the governing equations of coupled fields (see thermo-elastic coupling). 'vVith Onsager's relations we were able to obtain numerical values of coefficients for practical applications (see com- posite materials).

References

[21j TRUESDELL, C.: Rational Thermodynamics. McGraw-Hill, 1969.

[ GYARMATI, I.: Non-Equilibrium Thermodynamics (in Hungarian) Miiszaki Konyvkiad6, 1976.

[3] PRIGOGINE, I. - STENGERS, 1.: La nouvelle alliance. Metamorphose de la science.

Gallimard, Paris, 1986.

[4] LAITINEN, M. - PRAl\lILA, A. - SZEKERES, A.: Extrema of Coefficients of Ther- mal/Moisture Expansion for Thermo-Hygro-Elastic Composites. SES 95. Oct. 29.- Nov. 2. 1995, New Orleans, LA.

[5] SZEKERES, A. - ENGELBRECHT, J. (1994): Coupled Thermal and Moisture Fields with Application to Composites. Research Report Mech 108/94 Inst. of Cybernetics, Estonian Academy of Sciences, Tallin.

[6] SZEKERES, A. - HELLER, R.: (1994): Basic Equations of Hygrc)-Thelrm,o-l\1ec:hanicaJ:1 Materials and Application to Composites. Presentation at 2nd Thermal tures Conference, Oct. 18-20,1994, Univ of Virginia, Charlottesville, Virginia.

[7] SZEKERES A. - HELLER, R. (1995): Special Problems of Thermo-Hygro-Elastic pling. Presentation held at CANCAM 95, May 28-June 2, 1995, Victoria, B. C.